This invention relates to the representation of information representative of image data, particularly line-drawn graphics. In particular, the invention relates to using parametric trigonometric polynomials to represent curves.
There are many techniques for representing line-drawn graphics, such as curves, in compressed form. One such method involves the selection of a pair of polynomials for the approximated parametric representation of a given planar curve. Generally speaking, the higher the order, the better the approximation for a given curve.
A general theoretical background on the abstract problems associated with the selection of the most suitable pair of the polynomials for representing a given curve is presented by Blagovest Sendov, Theory of Approximation of Functions 322-29 (Moscow, Nauka 1977) (in Russian), and/or "Some Problems of the Theory of Approximation of Functions and Sets in Hausdorff Metric," Usp. Mat. Nauk, 24, No. 5(149), pp. 141-78 (1969).
A theoretical discussion of a method for finding a serviceable pair of polynomials using an iterative transformation-reparametrization process can be found in the 1983 publication by V. G. Polyakov et al., "Choice of Parametric Representation in Numerical Approximation and Encoding of Curves," (Translated from 20:3 Problemy Peredachi Informatsii 47-58, July-Sept., 1984), incorporated herein by reference. It is presumed here that those of ordinary skill in the relevant art are familiar with the aforementioned Polyakov article.
For complicated curves, the process contemplated by the prior Polyakov paper takes a relatively long time to complete approximations due to low convergence and the high degree of precision needed for computation. Furthermore, some applications require real-time approximation of the trajectory curves. For example, applications such as telewriting require on-line approximation of handwriting and drawing as the pen moves over the pad. In these situations, piecewise approximation is desirable. However, it is difficult to obtain good quality approximation and efficient representation using known piecewise approximation approaches such as splines.
What is desired is an improved compression-oriented piecewise polynomial approximation method which can employ the advantages of the iterative transformation-reparametrization technique in order to improve the compression efficiency for complicated curves.